Exponential Regression Calculator
Paste x,y data (one pair per line) and this calculator fits the best-fitting exponential model y = a·e^(bx) using least-squares linearisation. It returns the equation, the R² goodness-of-fit, and a predicted value at any x you choose.
Proportion of variance in y explained by the model (1.0 = perfect fit)
- 1
b = (n·ΣxlnY − Σx·ΣlnY) ÷ (n·Σx² − (Σx)²)
(5 × 29.9985 − 10 × 9.9955) ÷ 50 = 1.0008Slope from least-squares on (x, ln y). - 2
a = e^((ΣlnY − b·Σx) ÷ n)
e^((9.9955 − 1.0008 × 10) ÷ 5) = 0.9976 - 3
SS_tot = Σ(yᵢ − ȳ)²
1,975.892Total sum of squares around the mean y. - 4
SS_res = Σ(yᵢ − ŷᵢ)²
0.0018 - 5
R² = 1 − SS_res ÷ SS_tot
1 − 0.0018 ÷ 1,975.892 = 1
How does this calculator work?
Exponential regression fits y = a·e^(bx) by linearising to ln(y) = ln(a) + bx and applying least squares. Enter x,y data pairs — one per line — to get the equation, R² goodness-of-fit, and a prediction at any x. Only rows with y > 0 are used.
Formula
How this is calculated
Exponential regression finds the values of a and b that minimise the sum of squared residuals for the model y = a·e^(bx). Direct nonlinear optimisation is complex, so the standard approach linearises the problem: taking the natural logarithm of both sides gives ln(y) = ln(a) + b·x, which is a straight line in (x, ln(y)) space. Ordinary least-squares linear regression on (x, ln(y)) gives b (the slope) and ln(a) (the intercept), from which a = e^(intercept).
This method requires y > 0 for all data points — the logarithm is undefined otherwise. Rows with y ≤ 0 are silently excluded. The linearisation means the regression minimises squared errors in ln(y) space, not in y space. This weights smaller y-values more heavily. For data with large y-values and small relative errors, the results are excellent; for data with large absolute noise at small y, a nonlinear solver would be more appropriate.
R² (coefficient of determination) is reported on the original y scale: R² = 1 − SS_res/SS_tot. A value close to 1 indicates a strong exponential fit. A value much below 0.9 suggests the exponential model may not be appropriate for the data.
Frequently asked questions
Exponential regression linearises the model by taking ln(y). The natural logarithm is only defined for positive numbers. Data points with y ≤ 0 cannot be fitted by an exponential model of the form a·e^(bx) anyway (which is always positive for any x).
The coefficient b is the continuous growth rate per unit of x. A positive b means the model grows exponentially; a negative b means exponential decay. The doubling time is ln(2)/b for growth, and the half-life is ln(2)/|b| for decay.
Use exponential regression when your data grows (or decays) by a roughly constant percentage per unit of x rather than a constant absolute amount. Scatter plots that curve upward steeply, or data spanning several orders of magnitude, are classic candidates. A straight line on a log-y vs x plot confirms exponential behaviour.
Also known as
TG we-Calculate Editorial Team. (2026). Exponential Regression Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/exponential-regression-calculator
TG we-Calculate Editorial Team. "Exponential Regression Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/exponential-regression-calculator.
TG we-Calculate Editorial Team, "Exponential Regression Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/exponential-regression-calculator
@misc{wecalculate_exponential_regression_calculator, title = {Exponential Regression Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/exponential-regression-calculator}}, year = {2026}, note = {TG we-Calculate} }
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